What is the strongest condition you can up with for \(\alpha\) such that the field extension over \(F\) is equal to the ring of polynomials over \(F\). Or in other words \(F(\alpha)=F[\alpha]\)?

The strongest I could come up with is that \(\alpha\) generates a group such that \(\langle\alpha\rangle\cong C_n\) where \(C_n\) is the \(n^\text{th}\) cyclic group of arbitrary order.

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TopNewestThe strongest I could come up with is that \(\alpha\) generates a group such that \(\langle\alpha\rangle\cong C_n\) where \(C_n\) is the \(n^\text{th}\) cyclic group of arbitrary order.

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I have gone further and said that \(\alpha\) has to be algebraic over \(F\).

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